・ Rays Snell’s Law Structure of the Earth

・ Seismic Waves Near-Field Terms (Static Displacements) Far-Field Terms (P, S, Surface waves)

・ Normal modes Free oscillations of the Earth

Theoretical Seismology 2: Wave Propagation

Seismic waves

Faulting

Homogeneous Earth

Structure in the Earth 　

Crust-MantleCore-Mantle

440 km660 km

Snell’s LawFermat’s Principle

1

2 sin 1 / sin 2 = n21

Air

Water

Rays

1

２

1 < 2

Ray Paths in a Layered Medium

1

２

1 > 2

Ｓ lower

Faster

Faster

Slower

1

2

3

Ray Paths in a Layered Medium

1/1

1/2

1/3

Distance

Time

Andrija Mohorovicic (1857-1936)

Found seismic discontinuity at 30 km depth in the Kupa Valley (Croatia).

Mohorovicic discontinuity or ‘Moho’ 　

Boundary between crust and mantle

Moho

Forward Branch

Backward Branch

Forward Branch

Backward Branch

Forward Branch

Shadow Zone

Forward Branch

Backward Branch

Forward Branch

Shadow Zone

PcP

・ 1912 Gutenberg observed shadow zone 105o to 143o

・ 1939 Jeffreys fixed depth of core at 2898 km (using PcP)

ForwardBranch

BackwardBranch

ForwardBranch

PPcP

PKP

Shadow Zone

PcP

Core Reflections

P Mantle PS Mantle SK Outer core

PI Inner core

Pc Reflection

from the outer core

i Reflection from the inner core

diff Diffracted arrival

Aspects of Waves not Explained by Ray Theory

・ Different types of waves (P, S) ・ Surface Waves ・ Static Displacements ・ Frequency content

Seismic Waves

Wave Equation　

21

2

21

12 1

tu

cxu

1-D wave equation　 c = propagation speed　　　

Slinky: constant velocity wave propagation, no mass transfer, different from circulation eq.

1-D Wave Equation 　21

2

21

12 1

tu

cxu

LW 3.2.1

2

T T = wave period 　 = angular frequency 　

)]/(sin[),( cxtAtxu

Solution

Wave Period and Wavelength 　

wavelength 300 km

Velocity 6 km/sx

t

wavelength　　

period

Space

Time

period 50 sfrequency = 1/period= 0.02 hz

Velocity = Wavelength / Period

Body waves （ P ・ S ） 0.01 to 50 sec 50 m to 500 km

Surface waves 10 to 350 sec 30 to 1000 km

Free Oscillations 350 to 3600 sec 1000 to 10000 km

Static Displacements -

Period Wavelength

)()()2(2

2

uuftu

3-D Wave Equation with Source　source spatial 2nd derivative

Solution

dtM

rAtxu

r

r

N )(14

1),(/

/ 04 )(14

1)(14

1022022

rtMr

ArtMr

A ISIP

)(14

1)(14

10303

rtMr

ArtMr

A FSFP

Near-field Terms (Static Displacements)

Far-field Terms (P, S Waves)

Near-field terms

・ Static displacements

・ Only significant close to the fault 　　

・ Source of tsunamis　

r/ r/

t 　→r/ r/

Static Displacements

Bei-Fung Bridge near Fung-Yan city, 1999 Chi-Chi, Taiwan earthquake

Static displacements

Co-seismic deformationof 2003 Tokachi-okiEarthquake (M8.0)

Generation of Tsunami from Near-field Term

Far-field Terms

・ Propagating Waves

・ No net displacement 　

・ P waves

・ S waves

　　　　　　　　　　　　　　　　　　　　　　

)(14

1)(14

10303

rtMr

ArtMr

A FSFP

surface waves

Surface Waves

S

Shearer, Fig. 8.1

Period (sec)

Love

RayleighGr

oup

　Velo

city

(km

/sec

)

January 26, 2001 Gujarat, India Earthquake (Mw7.7)

Recorded in Japan at a distance of 57o (6300 km)

Love Waves

vertical

radial

transverse

Rayleigh Waves

Amplitude and IntensitySeismic waves loose amplitude with distance traveled - attenuation A(t) = A0e -ω0t/2Q

So the amplitude of the waves depends on distance from the earthquake. Therefore unlike magnitude intensity is not a single number.

I Barely felt II Felt by only few people III Felt noticeably, standing autos rock slightlyIV Felt by many, windows and walls creak V Felt by nearly everyone, some dished and windows brokenVI Felt by all, damaged plaster and chimneysVII Damage to poorly constructed buildingsVIII Collapse of poorly constructed buildings,

slight damage to well built structuresIX Considerable damage to well constructed buildings, buildings shifted off foundationsX Damage to well built wooden structures, some masonary buildings destroyed, train rails bent, landslides XI Few masonary structure remain standing, bridges destroyed, ground fissuresXII Damage total

Modified Mercalli Intensity

Normal Modes Normal Modes

(Daishinji, Fukui Prefecture)(Daishinji, Fukui Prefecture)Free Oscillations of the Earth 1960 Chile Earthquake

Useful for studies of ・ Interior of the Earth ・ Largest earthquakes

(Stein and Gellar 1978)

Toroidal and Spheroidal Modes

ToroidalSpheroidal

Dahlen and Tromp Fig. 8.5, 8.17

Natural Vibrations of the Earth

Shearer Ch.8.6Shearer Ch.8.6Lay and Wallace, Ch. 4.6Lay and Wallace, Ch. 4.6

Houseman http://earth.leeds.ac.uk/~greg/?Sphar/index.html

Free Oscillations l=1 m=1

Houseman http://earth.leeds.ac.uk/~greg/?Sphar/index.html

Free Oscillations l=1 m=2

Houseman http://earth.leeds.ac.uk/~greg/?Sphar/index.html

Free Oscillations l=1 m=3

Summary

Rays Earth structure causes complicated ray paths through the Earth (P, PKP, PcP)

Wave theory explains ・ P and S waves ・ Static displacements ・ Surface waves

Normal Modes The Earth rings like a bell at long periods